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Magnetic properties of the spinel systems ACr2X4 (A = Zn, Cd, Hg; X = S, Se)
Journal of Alloys and Compounds 489 (2010) 441–444
Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jallcom
Review
Magnetic properties of the spinel systems ACr2 X4 (A = Zn, Cd, Hg; X = S, Se) R. Masrour Laboratoire de Physique du Solide, Université Sidi Mohammed Ben Abdellah, Faculté des sciences Dhar Mahraz, BP 1796, Fez, Morocco
a r t i c l e
i n f o
Article history: Received 15 August 2008 Accepted 23 September 2009 Available online 2 October 2009 PACS: 71.45.Gm 77.80.Bh 75.40.Cx Keywords: Spinel Critical temperature Exchange interactions High-temperature series expansions Padé approximants Critical exponents Phase diagram
a b s t r a c t The magnetic properties of the spinels systems Zn1−x Cdx Cr2 Se4 and Hg1−x Cdx Cr2 S4 have been studied by mean-field theory and high-temperature series expansions in the range 0 ≤ x ≤ 1. The nearest neighbour and the next-nearest-neighbour super-exchange interaction J1 (x) and J2 (x) respectively, are calculated for the two systems, using the first theory in the range 0 ≤ x ≤ 1. The intra-planar and the inter-planar interactions are deduced. The corresponding classical exchange energy for the magnetic structures is obtained. The second theory has been applied to the spinel systems Zn1−x Cdx Cr2 Se4 and Hg1−x Cdx Cr2 S4 , combined with the Padé approximants method, to determine the magnetic phase diagrams (TC versus dilution x) in the range 0 ≤ x ≤ 1. The obtained theoretical results are in agreement with the experimental ones obtained by magnetic measurements. The critical exponents associated with the magnetic susceptibility () and with the correlation lengths (v) are deduced for the two systems in the ordered phase. © 2009 Elsevier B.V. All rights reserved.
Contents 1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High-temperature series expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussions and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction During the last decade the fascinating physics of spinel compounds came into the focus of modern solid-state physics and materials science. Materials with spinel structures, with the formula AB2 X4 , are of continuing interest because of their wide variety of physical properties. This is essentially related to (i) the existence of two types of crystallographic sublattices, tetrahedral (A) and octahedral (B), available for the metal ions; (ii) the great flexibility of the structure in hosting various metal ions, differently distributed over the two sublattices, with a large possibility of reciprocal substitution between them. Solid solutions of thiospinels and selenospinels have received considerable attention because of their interesting electrical and magnetic properties, which can
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441 442 442 442 443
vary greatly as a function of composition [1–7]. The compounds HgCr2 S4 , CdCr2 S4 , HgCr2 Se4 and CdCr2 Se4 are known to be ferromagnetic spinels [8–11], whereas ZnCr2 S4 and ZnCr2 Se4 are known to be antiferromagnetic [12,13]. The chromium ion Cr3+ occupies the octahedral sites in the spinel B sublattice and the Cd2+ occupies the tetrahedrally coordinated A sublattice. In the mixed spinels Zn1−x Cdx Cr2 Se4 and Hg1−x Cdx Cr2 S4 , two important frustration effects are present: the topological frustration in presence of antiferromagnetic interactions and the competition between n.n. ferromagnetic Cr–Se(S)–Cr interactions J1 and high-order-neighbours antiferromagnetic Cr–Se(S)–(Zn, Cd, Hg)–Se(S)–Cr interactions J2 . Zn, Hg and Cd are involved as an intermediate in super-exchange interactions between higher-order neighbours. In this work, the first- and the second-nearest-neighbours exchange interactions J1 (x) and J2 (x) are calculated on the basis of magnetic-measurement results in Zn1−x Cdx Cr2 Se4 and
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R. Masrour / Journal of Alloys and Compounds 489 (2010) 441–444
Table 1 The critical temperature TC (K), the Néel temperature TN (K), the Curie–Weiss temperature p , values of the first, second, intra-plane, inter-plane exchange integrals, the ratio of inter- to-intra-planar and the energy of Zn1−x Cdx Cr2 Se4 as a function of dilution x. x
TN (K) [14]
TC (K) [14]
p (K) [14]
J1 /kB (K)
J2 /kB (K)
Jaa /kB (K)
Jab /kB(K)
Jac /kB (K)
|(Jab + Jac )/Jaa |
E/kBS2 (K)
1.0 0.8 0.6
– – –
130.00 103.50 72.60
204.00 195.13 174.30
19.800 16.854 13.070
−3.100 −1.922 −0.725
39.600 33.708 26.140
54.400 52.040 46.480
−12.400 −7.688 −2.900
1.060 1.315 1.667
81.600 78.060 69.720
Hg1−x Cdx Cr2 S4 in the range 0 ≤ x ≤ 1 [14]. The values of the intraplane and inter-plane interactions Jaa , Jab and Jac , respectively, are deduced. The interaction energy of the magnetic structure for the two systems is obtained, using the values of J1 (x) and J2 (x) obtained in the range 0 ≤ x ≤ 1. In recent work [1,2], we have used the high-temperature series expansions (HTSE) to study the thermal and disorder variation of the short-range order (SRO) in the particular B-spinel compound ZnCr2x Al2−2x S4 . The Padé approximant (P.A.) [15] analysis of the HTSE of the correlation length has been shown to be a useful method for the study of the critical region [16–19]. The model that we used in the present work is valid in the case of a spinel structure. We have used the HTSE technique to determine the critical temperature TC or the Néel temperature TN and the critical exponent and v associated with the magnetic susceptibility and with correlation length , respectively, in the ordered phase.
where (ϕ) is the eigenvalue of the matrix formed by the Fourier transform of the exchange integral. Using Eqs. (2)–(4) together with the experimental values of TC , TN and p [14], the values of J1 (x) and J2 (x) have been determined for each composition of the system. The optimum values are given in Tables 1 and 2. In the same tables we give also the values of the intra-plane and inter-plane interactions Jaa = 2J1 ; Jab = 4J1 + 8J2 and Jac = 4J2 ; respectively and the ratio of inter- to intra-planar interactions Jinter /Jintra = |(Jab + Jac )/Jaa | for 0 ≤ x ≤ 1. The obtained values of the exchange integrals (J1 and J2 ) will be used in the following section. The values of the corresponding classical exchange energy E/kBS2 = 6J1 + 12J2 [21] for the magnetic structure are given.
2. Theoretical method
3. High-temperature series expansions
Starting with the well-known Heisenberg model, the Hamiltonian of the system is given by:
In this section we have used the results given by the HTSE for both the zero field magnetic susceptibility and the correlation length (T) with arbitrary y = J2 /J1 up to sixth order ˇ in Ref [22]. Figs. 1 and 2, show magnetic phase diagrams of Zn1−x Cdx Cr2 Se4 and Hg1−x Cdx Cr2 S4 systems, respectively. We can see the good agreement between the magnetic phase diagrams obtained by the HTSE technique and the experimental ones [23–26]. The simplest assumption that one can make concerning the nature of the singularity of the magnetic susceptibility (T) is that at the neighbourhood of the critical point the above two functions exhibit an asymptotic behaviour:
H = −2
Jij Si Sj
(1)
i,j
where Jij is the exchange integral between the spins situated at sites
i and j. Si is the atomic spin of the magnetic ion located on the ith site. The factor “2” in Eq. (1) arises from the fact that, when summing over all possible pairs ij exchange interactions, we count each pair twice. The mean-field approximation leads to simple relations between the paramagnetic Curie-temperature p and the critical temperature TC and the two considered exchange integrals J1 and J2 . In the case of spinels containing the magnetic moment only in the octahedral sublattice, the mean-field approximation of this expression leads to simple relations between the paramagnetic Curie-temperature p and the critical temperature TC ; respectively, and the two exchange integrals J1 and J2 : these can be used to derive numerical values for the exchange constants. Following the method of Holland and Brown [20], the expression of TC and p that can describe the systems Zn1−x Cdx Cr2 Se4 and Hg1−x Cdx Cr2 S4 are: TC =
2S(S + 1) [2J1 − 4J2 ] 3kB
(2)
P =
2S(S + 1) [6J1 + 12J2 ] 3kB
(3)
In the antiferromagnetic region, the Néel temperature TN is given by [21]: TN =
5 (ϕ) 2kB
(4)
(T ) ∝ (T − TC )−
(3’)
2 (T ) ∝ (TC − T )−2
(4’)
Estimates of TC or TN , and v for Zn1−x Cdx Cr2 Se4 and Hg1−x Cdx Cr2 S4 have been obtained using the Padé approximate method (P.A.) [15]. The [M, N] PA to the series F(ˇ) is a rational fraction PM /QN , with PM and QN , polynomials of degree M and N in ˇ = 1/kBT, satisfying: F(ˇ) = PM /QN + O(ˇM+N+1 ). The sequence of [M, N] P.A. to be both log((T)) and log( 2 (T)) are found to be convergent. The simple pole corresponds to TC or TN and the residues to the critical exponents and v. 4. Discussions and conclusions J1 (x) and J2 (x) have been determined from mean-field theory, using the experimental data of TC or TN and p [14] for each
where kB is the Boltzmann constant and SCr = 3/2.
Table 2 The critical temperature TC (K), the Néel temperature TN (K), the Curie–Weiss temperature p , values of the first, second, intra-plane, inter-plane exchange integrals, the ratio of inter- to-intra-planar and the energy of Hg1−x Cdx Cr2 S4 as a function of dilution x. x
TN (K) [14]
TC (K) [14]
p (K) [14]
J1 /kB (K)
J2 /kB (K)
Jaa /kB (K)
Jab /kB (K)
Jac /kB (K)
|(Jab + Jac )/Jaa |
E/kBS2 (K)
1.0 0.6 0.4
– – –
84.00 71.32 61.88
152.78 159.86 158.68
13.492 12.46 11.477
−1.653 −0.901 −0.449
26.984 24.920 22.954
40.744 42.632 42.316
−6.612 −3.604 −1.796
1.246 1.566 1.765
61.116 63.948 63.474
R. Masrour / Journal of Alloys and Compounds 489 (2010) 441–444
Fig. 1. Magnetic phase diagram of Zn1−x Cdx Cr2 Se4 . The various phases are the paramagnetic phase (PM), ferromagnetic phase (FM) (0.5 ≤ x ≤ 1), mixed phase (0.4 ≤ x < 0.5) and helimagnetic (HM) (0 ≤ x < 0.4). The solid circles are the present results (given by HTSE method). The solid star represents the experimental points deduced by magnetic measurements [14].
443
decreases. This variation is responsible on the apparition of the magnetic spiral structure (HM) in ZnCr2 Se4 . From Table 2 corresponding to Hg1−x Cdx Cr2 S4 , the situation is different. J1 (x), J2 (x) and the exchange energy decrease in absolute value with deceases x. It is found that the intra-plane interaction Jaa at high concentrations of Cadmium, but the inter-plane interaction appears and reduces the stabilization of ferromagnetism at sufficiently low concentrations. The HTSE extrapolated with Padé approximants method is shown to be a convenient method to provide valid estimations of the critical temperatures for real system. By applying this method to the magnetic susceptibility (T) we have estimated the critical temperature TC or TN for each dilution x. The obtained magnetic phase diagrams of the Zn1−x Cdx Cr2 Se4 and Hg1−x Cdx Cr2 S4 systems are presented in Figs. 1 and 2, respectively. Several thermodynamic phases may appear including the paramagnetic (PM), ferromagnetic (FM) 0.5 ≤ x ≤ 1 for Zn1−x Cdx Cr2 Se4 (0.4 ≤ x < 1 for Hg1−x Cdx Cr2 S4 ), mixed phases for Zn1−x Cdx Cr2 Se4 (0.4 ≤ x < 0.5) and for Hg1−x Cdx Cr2 S4 ) in 0.35 ≤ x < 0.4, helimagnetic (HM) in 0 ≤ x ≤ 03.5 for Hg1−x Cdx Cr2 S4 and in 0 ≤ x ≤ 0.4 for Zn1−x Cdx Cr2 Se4 . In these figures we have included, for comparison, the experimental results obtained by magnetic measurement. From these figures one can see good agreement between the theoretical phase diagram and experimental results. In addition, we have determined the region spin glass while using the expression of the nonlinear susceptibility. In the other hand, the value of critical exponents and v associated with the magnetic susceptibility (T) and with the correlation length (T), have been estimated in the ordering phase. The sequence of [M, N] PA to series of (T) has been evaluated. By examining the behaviour of these PA, the convergence was found to be quite rapid. Estimates central values of the critical exponents associated with the magnetic susceptibility and with the correlation length are found to be = 1.373 ± 0.002, v = 0.695 ± 0.004 and = 1.381 ± 0.002, v = 0.694 ± 0.008, for two systems Zn1−x Cdx Cr2 Se4 and Hg1−x Cdx Cr2 S4 , respectively. These values are insensitive to dilution x in phase ordering for two systems. The values of and are nearest to the one of 3D Heisenberg model [27], namely, 1.3866 ± 0.0012, and 0.7054 ± 0.0011. To conclude, it would be interesting to compare the critical exponents with other theoretical values. A lot of methods of extracting critical exponents have been given in the literature. We have selected many of the methods, and summarised our findings below. Zarek [28] has found experimentally by magnetic balance for CdCr2 Se4 is = 1.29 ± 0.02; for HgCr2 Se4 = 1.30 ± 0.02 and for CuCr2 Se4 is = 1.32 ± 0.02. References
Fig. 2. Magnetic phase diagram of Hg1−x Cdx Cr2 S4 . The various phases are the paramagnetic phase (PM), ferromagnetic phase (FM) (0.4 ≤ x < 1), phase mixed (0.35 ≤ x < 0.4), and helimagnetic (HM) phase (0 ≤ x ≤ 03.5). The solid circles are the present results (given by HTSE method). The solid star represents the experimental points deduced by magnetic measurements [14].
dilution. From these values we have deduced the values of the intraplane and inter-plane interactions Jaa , Jbb and Jac , respectively, the energy of the magnetic structures and the ratio of inter to intraplanar interactions Jinter /Jintra = |(Jab + Jac )/Jaa |. The values obtained for Zn1−x Cdx Cr2 Se4 and Hg1−x Cdx Cr2 S4 are given in Tables 1 and 2 in the range 0 ≤ x ≤ 1. From Table 1 corresponding to Zn1−x Cdx Cr2 Se4 , on can see that J1 (x) and the absolute value of J2 (x) decreases with x. The decrease in J1 (x) is due to the change in the average value of the Cr–Cr distance. J2 (x) represents the strength of the Cr–Se–A–Se–Cr superexchange, we expect its value to depend on the nature of the A-site cation. The ratio Jinter /Jintra = |(Jab + Jac )/Jaa | increases when x
[1] M. Hamedoun, M. Houssa, N. Benzakour, A. Hourmatallah, J. Phys: Condens. Matter 10 (1998) 3611. [2] M. Hamedoun, M. Houssa, N. Benzakour, A. Hourmatallah, Physica B 270 (1999) 384. [3] Y. Cherriet, M. Hamedoun, J. Magn. Magn. Mater. 224 (2001) 106; F. Mahjoubi, M. Hamedoun, F.Z. Bakkali, A. Hourmatallah, A. Benyoussef, J. Magn. Magn. Mater. 221 (2000) 359. [4] A.I. Abramovich, L.I. Koreleva, L.N. Lukina, Phys. Solid State 41 (1999) 73. [5] A. Rachadi, M. Hamedoun, D. Allam, Physica B 222 (1996) 160. [6] J. Hemberger, H.A. Krug von Nidda, V. Tsurkan, A. Loidl, Phys. Rev. Lett. 97 (2006) (id 087204). [7] E. Loudghiri, A. Belayachi, M. Nogues, M. Taibi, M.M. Cruz, M. Godinho, J. Magn. Magn. Mater. 320 (2008) 671–682. [8] P.K. Baltzer, H.W. Lehmann, M. Robbins, Phys. Rev. Lett. 15 (1965) 493. [9] H.W. Lehmann, M. Robbins, J. Appl. Phys. 37 (1966) 1389. [10] N. Menyuk, K. Dwight, R.J. Arnot, A. Wold, J. Appl. Phys. 37 (1966) 1387. [11] P.K. Baltzer, P.J. Wojtowicz, M. Robbins, E. Lopatin, Phys. Rev. 151 (1966) 367. [12] P.K. Lotgering, Proceeding of International Conference on Magnetism Nottingham England 1964, Institute of Physics and Physical Society, London, 1965, p. 533. [13] R. Plumier, Compt. Rend 260 (1966) 3348. [14] P.K. Baltzer, M. Robbins, P.J. Wojtowicz, J. Appl. Phys. 38 (1967) 3. [15] “Padé Approximants”, edited by G.A. Baker, P. Graves-Morris, Addison-Wesley, London, 1981.
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[16] R. Navaro, in: L.J.D.E. Jonsgh (Ed.), Magnetic Properties of Layered Transition Metal Compounds, 105, Kluwer, Daventa, 1990. [17] M.C. Moron, J. Phys: Condens. Matter 8 (1996) 11141. [18] H. Bakrim, K. Bouslykhane, M. Hamedoun, A. Hourmatallah, N. Benzakour, J. Magn. Magn. Mater. 285 (2005) 327–334. [19] M. Hamedoun, R. Masrour, K. Bouslykhane, A. Hourmatallah, N. Benzakour, J. Alloys Compd. 462 (2008) 125–128. [20] W.E. Holland, H.A. Brown, Phys-Status Sol. (a) 10 (1972) 249. [21] M. Hamedoun, A. Wiedenmann, J.L. Dormann, M. Nogues, J. Rossat-Mignod, J. Phys. C: Solid State Phys. 19 (1986) 1783–1800. [22] N. Benzakour, M. Hamedoun, M. Houssa, A. Hourmatallah, F. Mahjoubi, Phys. Status Sol. (b) 212 (1999) 335.
[23] M. Alba, M. Hammann, J. Nougues, J. Phys., C 15 (1982) 5441. [24] K. Afif, A. Benyoussef, M. Hamedoun, A. Hourmatallah, Phys. Status Sol. (b) 219 (2000) 383. [25] M. Hamedoun, R. Masrour, K. Bouslykhane, A. Hourmatallah, N. Benzakour, Phys. Status Sol. (c) 3 (2006) 3307–3310. [26] M. Hamedoun, K. Bouslykhane, R. Masrour, F. Talbi, A. Hourmatallah, N. Benzakour, M.J. Condensed, Matter 8 (2007) 1. [27] For a review, see M.F. Collins, Magnetic Critical Scattering, Oxford University Press, Oxford, 1989. [28] W. Zarek, Acta Phys. Polon. A 52 (1977) 657.
Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jallcom
Review
Magnetic properties of the spinel systems ACr2 X4 (A = Zn, Cd, Hg; X = S, Se) R. Masrour Laboratoire de Physique du Solide, Université Sidi Mohammed Ben Abdellah, Faculté des sciences Dhar Mahraz, BP 1796, Fez, Morocco
a r t i c l e
i n f o
Article history: Received 15 August 2008 Accepted 23 September 2009 Available online 2 October 2009 PACS: 71.45.Gm 77.80.Bh 75.40.Cx Keywords: Spinel Critical temperature Exchange interactions High-temperature series expansions Padé approximants Critical exponents Phase diagram
a b s t r a c t The magnetic properties of the spinels systems Zn1−x Cdx Cr2 Se4 and Hg1−x Cdx Cr2 S4 have been studied by mean-field theory and high-temperature series expansions in the range 0 ≤ x ≤ 1. The nearest neighbour and the next-nearest-neighbour super-exchange interaction J1 (x) and J2 (x) respectively, are calculated for the two systems, using the first theory in the range 0 ≤ x ≤ 1. The intra-planar and the inter-planar interactions are deduced. The corresponding classical exchange energy for the magnetic structures is obtained. The second theory has been applied to the spinel systems Zn1−x Cdx Cr2 Se4 and Hg1−x Cdx Cr2 S4 , combined with the Padé approximants method, to determine the magnetic phase diagrams (TC versus dilution x) in the range 0 ≤ x ≤ 1. The obtained theoretical results are in agreement with the experimental ones obtained by magnetic measurements. The critical exponents associated with the magnetic susceptibility () and with the correlation lengths (v) are deduced for the two systems in the ordered phase. © 2009 Elsevier B.V. All rights reserved.
Contents 1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High-temperature series expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussions and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction During the last decade the fascinating physics of spinel compounds came into the focus of modern solid-state physics and materials science. Materials with spinel structures, with the formula AB2 X4 , are of continuing interest because of their wide variety of physical properties. This is essentially related to (i) the existence of two types of crystallographic sublattices, tetrahedral (A) and octahedral (B), available for the metal ions; (ii) the great flexibility of the structure in hosting various metal ions, differently distributed over the two sublattices, with a large possibility of reciprocal substitution between them. Solid solutions of thiospinels and selenospinels have received considerable attention because of their interesting electrical and magnetic properties, which can
E-mail address: [email protected]. 0925-8388/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2009.09.149
441 442 442 442 443
vary greatly as a function of composition [1–7]. The compounds HgCr2 S4 , CdCr2 S4 , HgCr2 Se4 and CdCr2 Se4 are known to be ferromagnetic spinels [8–11], whereas ZnCr2 S4 and ZnCr2 Se4 are known to be antiferromagnetic [12,13]. The chromium ion Cr3+ occupies the octahedral sites in the spinel B sublattice and the Cd2+ occupies the tetrahedrally coordinated A sublattice. In the mixed spinels Zn1−x Cdx Cr2 Se4 and Hg1−x Cdx Cr2 S4 , two important frustration effects are present: the topological frustration in presence of antiferromagnetic interactions and the competition between n.n. ferromagnetic Cr–Se(S)–Cr interactions J1 and high-order-neighbours antiferromagnetic Cr–Se(S)–(Zn, Cd, Hg)–Se(S)–Cr interactions J2 . Zn, Hg and Cd are involved as an intermediate in super-exchange interactions between higher-order neighbours. In this work, the first- and the second-nearest-neighbours exchange interactions J1 (x) and J2 (x) are calculated on the basis of magnetic-measurement results in Zn1−x Cdx Cr2 Se4 and
442
R. Masrour / Journal of Alloys and Compounds 489 (2010) 441–444
Table 1 The critical temperature TC (K), the Néel temperature TN (K), the Curie–Weiss temperature p , values of the first, second, intra-plane, inter-plane exchange integrals, the ratio of inter- to-intra-planar and the energy of Zn1−x Cdx Cr2 Se4 as a function of dilution x. x
TN (K) [14]
TC (K) [14]
p (K) [14]
J1 /kB (K)
J2 /kB (K)
Jaa /kB (K)
Jab /kB(K)
Jac /kB (K)
|(Jab + Jac )/Jaa |
E/kBS2 (K)
1.0 0.8 0.6
– – –
130.00 103.50 72.60
204.00 195.13 174.30
19.800 16.854 13.070
−3.100 −1.922 −0.725
39.600 33.708 26.140
54.400 52.040 46.480
−12.400 −7.688 −2.900
1.060 1.315 1.667
81.600 78.060 69.720
Hg1−x Cdx Cr2 S4 in the range 0 ≤ x ≤ 1 [14]. The values of the intraplane and inter-plane interactions Jaa , Jab and Jac , respectively, are deduced. The interaction energy of the magnetic structure for the two systems is obtained, using the values of J1 (x) and J2 (x) obtained in the range 0 ≤ x ≤ 1. In recent work [1,2], we have used the high-temperature series expansions (HTSE) to study the thermal and disorder variation of the short-range order (SRO) in the particular B-spinel compound ZnCr2x Al2−2x S4 . The Padé approximant (P.A.) [15] analysis of the HTSE of the correlation length has been shown to be a useful method for the study of the critical region [16–19]. The model that we used in the present work is valid in the case of a spinel structure. We have used the HTSE technique to determine the critical temperature TC or the Néel temperature TN and the critical exponent and v associated with the magnetic susceptibility and with correlation length , respectively, in the ordered phase.
where (ϕ) is the eigenvalue of the matrix formed by the Fourier transform of the exchange integral. Using Eqs. (2)–(4) together with the experimental values of TC , TN and p [14], the values of J1 (x) and J2 (x) have been determined for each composition of the system. The optimum values are given in Tables 1 and 2. In the same tables we give also the values of the intra-plane and inter-plane interactions Jaa = 2J1 ; Jab = 4J1 + 8J2 and Jac = 4J2 ; respectively and the ratio of inter- to intra-planar interactions Jinter /Jintra = |(Jab + Jac )/Jaa | for 0 ≤ x ≤ 1. The obtained values of the exchange integrals (J1 and J2 ) will be used in the following section. The values of the corresponding classical exchange energy E/kBS2 = 6J1 + 12J2 [21] for the magnetic structure are given.
2. Theoretical method
3. High-temperature series expansions
Starting with the well-known Heisenberg model, the Hamiltonian of the system is given by:
In this section we have used the results given by the HTSE for both the zero field magnetic susceptibility and the correlation length (T) with arbitrary y = J2 /J1 up to sixth order ˇ in Ref [22]. Figs. 1 and 2, show magnetic phase diagrams of Zn1−x Cdx Cr2 Se4 and Hg1−x Cdx Cr2 S4 systems, respectively. We can see the good agreement between the magnetic phase diagrams obtained by the HTSE technique and the experimental ones [23–26]. The simplest assumption that one can make concerning the nature of the singularity of the magnetic susceptibility (T) is that at the neighbourhood of the critical point the above two functions exhibit an asymptotic behaviour:
H = −2
Jij Si Sj
(1)
i,j
where Jij is the exchange integral between the spins situated at sites
i and j. Si is the atomic spin of the magnetic ion located on the ith site. The factor “2” in Eq. (1) arises from the fact that, when summing over all possible pairs ij exchange interactions, we count each pair twice. The mean-field approximation leads to simple relations between the paramagnetic Curie-temperature p and the critical temperature TC and the two considered exchange integrals J1 and J2 . In the case of spinels containing the magnetic moment only in the octahedral sublattice, the mean-field approximation of this expression leads to simple relations between the paramagnetic Curie-temperature p and the critical temperature TC ; respectively, and the two exchange integrals J1 and J2 : these can be used to derive numerical values for the exchange constants. Following the method of Holland and Brown [20], the expression of TC and p that can describe the systems Zn1−x Cdx Cr2 Se4 and Hg1−x Cdx Cr2 S4 are: TC =
2S(S + 1) [2J1 − 4J2 ] 3kB
(2)
P =
2S(S + 1) [6J1 + 12J2 ] 3kB
(3)
In the antiferromagnetic region, the Néel temperature TN is given by [21]: TN =
5 (ϕ) 2kB
(4)
(T ) ∝ (T − TC )−
(3’)
2 (T ) ∝ (TC − T )−2
(4’)
Estimates of TC or TN , and v for Zn1−x Cdx Cr2 Se4 and Hg1−x Cdx Cr2 S4 have been obtained using the Padé approximate method (P.A.) [15]. The [M, N] PA to the series F(ˇ) is a rational fraction PM /QN , with PM and QN , polynomials of degree M and N in ˇ = 1/kBT, satisfying: F(ˇ) = PM /QN + O(ˇM+N+1 ). The sequence of [M, N] P.A. to be both log((T)) and log( 2 (T)) are found to be convergent. The simple pole corresponds to TC or TN and the residues to the critical exponents and v. 4. Discussions and conclusions J1 (x) and J2 (x) have been determined from mean-field theory, using the experimental data of TC or TN and p [14] for each
where kB is the Boltzmann constant and SCr = 3/2.
Table 2 The critical temperature TC (K), the Néel temperature TN (K), the Curie–Weiss temperature p , values of the first, second, intra-plane, inter-plane exchange integrals, the ratio of inter- to-intra-planar and the energy of Hg1−x Cdx Cr2 S4 as a function of dilution x. x
TN (K) [14]
TC (K) [14]
p (K) [14]
J1 /kB (K)
J2 /kB (K)
Jaa /kB (K)
Jab /kB (K)
Jac /kB (K)
|(Jab + Jac )/Jaa |
E/kBS2 (K)
1.0 0.6 0.4
– – –
84.00 71.32 61.88
152.78 159.86 158.68
13.492 12.46 11.477
−1.653 −0.901 −0.449
26.984 24.920 22.954
40.744 42.632 42.316
−6.612 −3.604 −1.796
1.246 1.566 1.765
61.116 63.948 63.474
R. Masrour / Journal of Alloys and Compounds 489 (2010) 441–444
Fig. 1. Magnetic phase diagram of Zn1−x Cdx Cr2 Se4 . The various phases are the paramagnetic phase (PM), ferromagnetic phase (FM) (0.5 ≤ x ≤ 1), mixed phase (0.4 ≤ x < 0.5) and helimagnetic (HM) (0 ≤ x < 0.4). The solid circles are the present results (given by HTSE method). The solid star represents the experimental points deduced by magnetic measurements [14].
443
decreases. This variation is responsible on the apparition of the magnetic spiral structure (HM) in ZnCr2 Se4 . From Table 2 corresponding to Hg1−x Cdx Cr2 S4 , the situation is different. J1 (x), J2 (x) and the exchange energy decrease in absolute value with deceases x. It is found that the intra-plane interaction Jaa at high concentrations of Cadmium, but the inter-plane interaction appears and reduces the stabilization of ferromagnetism at sufficiently low concentrations. The HTSE extrapolated with Padé approximants method is shown to be a convenient method to provide valid estimations of the critical temperatures for real system. By applying this method to the magnetic susceptibility (T) we have estimated the critical temperature TC or TN for each dilution x. The obtained magnetic phase diagrams of the Zn1−x Cdx Cr2 Se4 and Hg1−x Cdx Cr2 S4 systems are presented in Figs. 1 and 2, respectively. Several thermodynamic phases may appear including the paramagnetic (PM), ferromagnetic (FM) 0.5 ≤ x ≤ 1 for Zn1−x Cdx Cr2 Se4 (0.4 ≤ x < 1 for Hg1−x Cdx Cr2 S4 ), mixed phases for Zn1−x Cdx Cr2 Se4 (0.4 ≤ x < 0.5) and for Hg1−x Cdx Cr2 S4 ) in 0.35 ≤ x < 0.4, helimagnetic (HM) in 0 ≤ x ≤ 03.5 for Hg1−x Cdx Cr2 S4 and in 0 ≤ x ≤ 0.4 for Zn1−x Cdx Cr2 Se4 . In these figures we have included, for comparison, the experimental results obtained by magnetic measurement. From these figures one can see good agreement between the theoretical phase diagram and experimental results. In addition, we have determined the region spin glass while using the expression of the nonlinear susceptibility. In the other hand, the value of critical exponents and v associated with the magnetic susceptibility (T) and with the correlation length (T), have been estimated in the ordering phase. The sequence of [M, N] PA to series of (T) has been evaluated. By examining the behaviour of these PA, the convergence was found to be quite rapid. Estimates central values of the critical exponents associated with the magnetic susceptibility and with the correlation length are found to be = 1.373 ± 0.002, v = 0.695 ± 0.004 and = 1.381 ± 0.002, v = 0.694 ± 0.008, for two systems Zn1−x Cdx Cr2 Se4 and Hg1−x Cdx Cr2 S4 , respectively. These values are insensitive to dilution x in phase ordering for two systems. The values of and are nearest to the one of 3D Heisenberg model [27], namely, 1.3866 ± 0.0012, and 0.7054 ± 0.0011. To conclude, it would be interesting to compare the critical exponents with other theoretical values. A lot of methods of extracting critical exponents have been given in the literature. We have selected many of the methods, and summarised our findings below. Zarek [28] has found experimentally by magnetic balance for CdCr2 Se4 is = 1.29 ± 0.02; for HgCr2 Se4 = 1.30 ± 0.02 and for CuCr2 Se4 is = 1.32 ± 0.02. References
Fig. 2. Magnetic phase diagram of Hg1−x Cdx Cr2 S4 . The various phases are the paramagnetic phase (PM), ferromagnetic phase (FM) (0.4 ≤ x < 1), phase mixed (0.35 ≤ x < 0.4), and helimagnetic (HM) phase (0 ≤ x ≤ 03.5). The solid circles are the present results (given by HTSE method). The solid star represents the experimental points deduced by magnetic measurements [14].
dilution. From these values we have deduced the values of the intraplane and inter-plane interactions Jaa , Jbb and Jac , respectively, the energy of the magnetic structures and the ratio of inter to intraplanar interactions Jinter /Jintra = |(Jab + Jac )/Jaa |. The values obtained for Zn1−x Cdx Cr2 Se4 and Hg1−x Cdx Cr2 S4 are given in Tables 1 and 2 in the range 0 ≤ x ≤ 1. From Table 1 corresponding to Zn1−x Cdx Cr2 Se4 , on can see that J1 (x) and the absolute value of J2 (x) decreases with x. The decrease in J1 (x) is due to the change in the average value of the Cr–Cr distance. J2 (x) represents the strength of the Cr–Se–A–Se–Cr superexchange, we expect its value to depend on the nature of the A-site cation. The ratio Jinter /Jintra = |(Jab + Jac )/Jaa | increases when x
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